Gcd-closed sets and divisibility among power LCM matrices

Abstract

Let $a,b$, and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $[S^a]$ the $n\times n$ matrix having the $a$th power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$-entry. For $x\in S$, let $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. In this article, we show that $[S^a]$ divides $[S^b]$ in the ring of $n\times n$ matrices over the integers if $a|b$ and $S$ is gcd closed (i.e., $\gcd(x_i,x_j)\in S$ for all integers $i$ and $j$ with $1\le i,j\le n$) and satisfies the condition $\mathcal G$ (that is, for any $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that ${\rm lcm}(y_1,y_2)=x$ and $\gcd(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in (2026, Bull. Aust. Math. Soc., 113, 231-243).